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Home , Projective space, Segre embedding

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I. INTRODUCTION

Quantum entanglement is at the heart of quantum physics, with crucial roles in quantum information theory, superdense coding and quantum teleportation among others. An important problem in entanglement theory is to obtain separability criteria. While there is a clear definition of separability, in general it is difficult to determine whether a given state is entangled or separable. A refinement of this problem is to quantify entanglement on a given entangled state. This is a broadly open problem, in the sense that there is not a unique established way of measuring entanglement, which might depend on the initial set-up and applications on has in mind. There is, however, a general consensus on the desirable properties of a good entanglement measure [1, 2]. Directly attached to measuring entanglement is the notion of maximally entangled state, central in teleportation protocols. Many different entanglement measures and the corresponding notions of maximal en- tanglement have been proposed. Methods range from using Bell inequalities, looking at inequalities in the larger of entanglement witnesses, spin squeezing inequalities, en- tropic inequalities, the measurement of nonlinear properties of the quantum state or the approximation of positive maps. We refer to the exhaustive reviews [2–4] for the basic aspects of entanglement including its history, characterization, measurement, classification and applications. A particularly simple description of separability of quantum states arises naturally in the setting of complex . In this setting, pure multiparticle states are identified with points in the complex projective PN , the set of lines of the complex space CN+1 that go through the origin. Entanglement is then understood via the categorical product of projective spaces: the . This is a map of complex algebraic varieties

1 (n) 1 2n 1 P P P − , (1) × ···× −→ whose image, called the Segre variety, is described in terms of a family of homogeneous quadratic equations in 2n variables, where n is the number of particles. The points of the Segre variety correspond precisely to separable states (see for instance [5, 6]). In this paper, we exploit this geometric viewpoint to show that the image of (1) is in fact given by the intersection of all Segre varieties defined via bipartite-type Segre maps

2ℓ 1 2n−ℓ 1 2n 1 P − P − P − , for all 1 ℓ n 1. (2) × −→ ≤ ≤ − Specifically, we show that a state is q-partite if and only if it lies in q of the images of Segre maps of the form (2). We do this after showing that the Segre embedding (1) may be decomposed in various equivalent ways, leading to a hypercube of dimension (n 1) whose edges are bipartite-type Segre maps. In this framework all the information about− the separability of the state in contained in the last applications (2) of the hypercube. There are numerous approaches to entanglement via Segre varieties that are related to the present work [7–13]. The hypercube viewpoint presented here is a novel approach that connects the notions of bipartite and q-partite in a geometric way. While the above results are extremely precise and intuitive, they are purely algebraic. In order to build a bridge from geometry to physics, we introduce a family of observables n,ℓ , with 1 ℓ n 1, which allow us to detect when a given n-particle state belongs to{J each} of the≤ images≤ of− the bipartite-type Segre maps (2). As a consequence, we obtain that a state is q-partite if and only if at least q of the observables n,ℓ vanish on this state, completely identifying the sub-partitions of the system. Each ofJ these observables are related to the trace of the squared reduced density matrix, also known as the bipartite non-extensive Tsallis entropy with entropic index two [14, 15]. They are always positive on 3 entangled states and our first applications indicate that they provide well-behaved and fine measures of entanglement. We briefly explain the contents of this paper. We begin with a warm-up Section II where we detail the theory and results for the well-understood settings of two- and three-particle states. In Section III we develop the geometric aspects of the paper. In particular, we describe the Segre hypercube and prove Theorem III.6 on geometric decomposability. The main result of Section IV is Theorem IV.3, where we match geometric decomposability with our family of observables related to non-extensive entropic measures. The two theorems are combined in Section V, where we study entanglement measures for pure multiparticle states 1 of spin- 2 and apply our observables on various well-known multiparticle states.

ACKNOWLEDGMENTS

We would like to thank Joan Carles Naranjo for his ideas in the proof of Lemma A.1.

J. Cirici would like to acknowledge partial support from the AEI/FEDER, UE (MTM2016- 76453-C2-2-P) and the Serra Húnter Program. J. Salvadó and J. Taron are partially supported by the Spanish grants FPA2016-76005-C2-1-PEU, PID2019-105614GB-C21, PID2019-108122GB-C32, by the Maria de Maeztu grant MDM-2014-0367 of ICCUB, and by the European INT projects FP10ITN ELUSIVES (H2020-MSCA-ITN-2015-674896) and INVISIBLES-PLUS (H2020-MSCA-RISE-2015-690575).

II. WARM-UP: TWO AND THREE PARTICLE STATES

We begin with the well-understood example of two particle entanglement. The initial set-up consists in two particles which can be shared between two different observers, A and B, that can perform quantum measures to each of the particles. A general pure state for 1 two spin- 2 particles can be written as ψ = z 00 + z 01 + z 10 + z 11 , (3) | iAB 0 | i 1 | i 2 | i 3 | i 2 where zi are complex numbers that satisfy the normalization condition zi = 1. The labels AB, which will be often omitted, indicate that A is acting on the first| particle| while B is acting on the second, so P

ij := i j , for i, j 0, 1 . | i | iA ⊗ | iB ∈{ } Entanglement is a property that can be inferred from statistical properties of different measurements by the observers of the system. In particular, we can compute the sum of the expected values of A measuring the spin of the state ψ in the different directions. With this idea in mind, we define: | i

3 2 A B(ψ) := 2 ψ σi I2 ψ , J ⊗ − | h | ⊗ | i | i=0 ! X where σi, for i =1, 2, 3, denote the Pauli matrices, σ0 = I2 is the identity matrix of size two and denotes the Kronecker product. Using the expression (3) for ψ we obtain ⊗ | i 2 A B(ψ)=4 z0z3 z1z2 . J ⊗ | − | 4

It is well-known that the state ψ is entangled if and only if | i z z z z =0. 0 3 − 1 2

Therefore we find that ψ is a product state if and only if A B (ψ)=0. When A B (ψ) > 0 we have an entangled| i state, which is considered to beJ maximal⊗ ly entangledJ when⊗ the observable reaches its maximum value at A B(ψ)=1. J ⊗ The measure given by A B(ψ) may also be interpreted in terms of the density matrix operator. Indeed, the densityJ ⊗ matrix ρ for a pure state ψ is given by | i

z0z0 z0z1 z0z2 z0z3 z z z z z z z z ρ = ψ ψ = 1 0 1 1 1 2 1 3 . | i h |  z2z0 z2z1 z2z2 z2z3   z3z0 z3z1 z3z2 z3z3    Compute the reduced matrix for the subsystem A by means of the partial trace on B,

z0z0 + z1z1 z0z2 + z1z3 ρA = TrBρ = . z2z0 + z3z1 z2z2 + z3z3   The trace in the subsystem A of the square reduced matrix gives

Trρ2 =1 2 z z z z 2. A − | 0 3 − 1 2| This magnitude is refereed in the literature as the Tsallis entropy or q-entropy with q = 2 and is directly related with the proposed measure by,

2 A B(ψ)=2 1 TrAρA . J ⊗ − Recall that by the Schmidt Decomposition, and after choosing a convenient , any pure 2-particle state may be written as

ψ = cos(θ) 00 + sin(θ) 11 | i | i | i where θ [0,π/4] is the Schmidt angle, which is known to quantify entanglement (see for instance∈ [16]). A simple computation gives

2 2 A B (ψ) = 4(cos (θ) sin (θ)). J ⊗ The two well-known states 1 Sep = 00 and EPS = ( 00 + 11 ), | i | i | i √2 | i | i which can be taken as representatives for the two only classes of states under the action of Stochastic Local Operations and Classical Communication (SLOCC), correspond to the two extreme cases of separated (θ =0) and maximally entangled (θ = π/4) states respectively, in the sense that EPS gives the greatest violation of Bell inequalities, has the largest entropy of entanglement,| andi its one-party reduced states are both maximally mixed. The characterization of entanglement has a simple geometric interpretation. Note first 1 that a general pure state for a single spin- 2 particle may be written as

z 0 + z 1 where z 2 + z 2 =1. 0 | i 1 | i | 0| | 1| 5

In particular, such a state is determined by the pair of complex numbers (z0,z1) and the normalization condition ensures (z0,z1) = (0, 0). This allows one to consider the corre- 6 1 sponding [z0 : z1] in the complex P . This space is defined 2 as the set of lines of C that go through the origin. The class [z0 : z1] denotes the the set C2 of all points (z0′ ,z1′ ) (0, 0) such that there is a non-zero λ with ∈ \{ } C (z0,z1)= λ(z0′ ,z1′ ). Note that, by construction, we have [z0 : z1] = [λz0 : λz1] for all λ ∗. 1 P1 ∈ In summary, every one particle state of spin- 2 defines a unique in . Conversely, given a point [z : z ] P1 we may choose a representative (z ,z ) such that z 2 + z 2 =1 0 1 ∈ 0 1 | 0| | 1| and so it determines a unique pure state z0 0 + z1 1 a global phase which does not affect any state measurements. | i | i This one-to-one correspondence between pure states and points in the 1 generalizes analogously to several particles: pure states for n particles of spin- 2 correspond 2n 1 to points in P − . This correspondence is just a way of describing the of the of quantum states and so is valid for particles of arbitrary spin, after adjusting accordingly. For our two-particle case, since the state ψ introduced in (3) is determined by the | i normalized set of complex numbers (z0,z1,z2,z3) we obtain a point [ψ] := [z0 : z1 : z2 : z3] in the P3, the set of lines in C4 going through the origin. Interestingly for the study of entanglement, there is a map

P1 P1 P3 fA B : A B AB, ⊗ × −→ called the Segre embedding, defined by the products of coordinates

[a : a ], [b : b ] [a b : a b : a b : a b ]. 0 1 0 1 7→ 0 0 0 1 1 0 1 1 The Segre map is the categorical product of projective spaces, describing how to take prod- ucts on projective Hilbert spaces. The word embedding accounts for the fact that this map is injective: it embeds the product P1 P1, which has complex dimension 2, inside P3, which has complex dimension 3. The image× of this map

ΣA B := Im(fA B) ⊗ ⊗ is called the Segre variety. This is a complex of dimension 2 and is given by the set of points [z : z : z : z ] P3 satisfying the single quadratic polynomial equation 0 1 2 3 ∈ z z z z =0. 0 3 − 1 2 In particular, product states correspond precisely to points in the Segre variety and

A B (ψ)=0 [ψ] ΣA B. J ⊗ ⇐⇒ ∈ ⊗ As a consequence, ψ is a product state if and only if its corresponding point [ψ] in P3 | iAB lies in the Segre variety ΣA B. ⊗ The entanglement of three particle states is also well-understood in the literature [17]. However, this case already exhibits some non-trivial facts that arise in the geometric inter- pretation of entanglement. We briefly review this case before discussing the general set-up. 1 A general pure state for three spin- 2 particles can be written as

ψ = z0 000 + z1 001 + z2 010 + z3 011 + | iABC | i | i | i | i . (4) +z 100 + z 101 + z 110 + z 111 4 | i 5 | i 6 | i 7 | i 6

As in the two particle case, zi are complex numbers that satisfy the normalization condition 2 zi =1. Now we have a third observer, C, in addition to A and B, so | | P ijk := i j k for i, j, k 0, 1 . | i | iA ⊗ | iB ⊗ | iC ∈{ } We will first measure bipartitions: the separability of this state into states of the form

ϕ ϕ′ or ϕ ϕ′ . | iA ⊗ | iBC | iAB ⊗ | iC For the first case, we define the observable

3 2 A BC (ψ) := 2 ψ σi I4 ψ = J ⊗ − | h | ⊗ | i | i=0 ! X =4 z z z z 2 + z z z z 2 + z z z z 2+ | 0 5 − 1 4| | 0 6 − 2 4| | 0 7 − 3 4| + z z z z 2 + z z z z 2 + z z z z 2 , | 1 6 − 2 5| | 1 7 − 3 5| | 2 7 − 3 6| where the last equality will be proven in Section IV. Let us for now interpret this expression geometrically. Our state ψ corresponds to a point in P7 and bipartitions of type A BC are geometrically characterized| i by the Segre embedding ⊗ P1 P3 P7 fA BC : A BC ABC ⊗ ⊗ −→ 7 defined by sending the [a0 : a1], [b0 : b1 : b2 : b3] to the point of P given by

[a0b0 : a0b1 : a0b2 : a0b3 : a1b0 : a1b1 : a1b2 : a1b3].

Specifically, the state ψ can be written as ϕ A ϕ′ BC if and only if the corresponding point [ψ] = [z : : z| ] i P7 lies in the Segre| varietyi ⊗ | i 0 ··· 7 ∈ ΣA BC := Im(fA BC). ⊗ ⊗ n The equations defining ΣA BC , having set coordinates [z0 : : z7] of P , are given by the vanishing of all 2 2 minors⊗ of the matrix ··· × z z z z 0 1 2 3 . z4 z5 z6 z7   Therefore we see that

A BC (ψ)=0 [ψ] ΣA BC . J ⊗ ⇐⇒ ∈ ⊗ For the second case, we define:

3 1 2 AB C (ψ) := 2 ψ σi σj I2 ψ = J ⊗ − 2 | h | ⊗ ⊗ | i |  i,j=0 X =4 z z z z 2+ z z z z 2 + z z z z 2+ | 0 3 − 1 2| | 0 5 − 1 4| | 0 7 − 1 6| + z z z z 2 + z z z z 2 + z z z z 2 , | 2 5 − 3 4| | 2 7 − 3 6| | 4 7 − 5 6| where again, the last equality is detailed in Section IV. Bipartitions of type AB C are now geometrically characterized by the Segre embedding ⊗ P3 P1 P7 fAB C : AB C ABC . ⊗ ⊗ −→ 7

The state ψ can be written as ϕ AB ϕ′ C if and only if the corresponding point [ψ]= [z : : z| ] i P7 lies in the Segre| varietyi ⊗ | i 0 ··· 7 ∈ ΣAB C := Im(fAB C). ⊗ ⊗ 7 In this case, the Segre variety ΣAB C is given by all points [z0 : : z7] P such that all 2 2 minors of the matrix ⊗ ··· ∈ × z0 z1 z2 z3 z4 z5 z6 z7   vanish. Therefore we may conclude that

AB C (ψ)=0 [ψ] ΣAB C . J ⊗ ⇐⇒ ∈ ⊗ We may now ask about separability of the state ψ in a totally decomposed form | i ϕ ϕ′ ϕ′′ . | iA ⊗ | iB ⊗ | iC It turns out that the above defined observables are sufficient in order to address this question. This is easily seen using the geometric characterization of entanglement, as we next explain. The total separability of the state ψ ABC is geometrically characterized by the generalized Segre embedding | i P1 P1 P1 P7 fA B C : A B C ABC ⊗ ⊗ × × −→ 7 defined by sending the tuples [a0 : a1], [b0 : b1], [c0 : c1] to the point in P given by

[a0b0c0 : a0b0c1 : a0b1c0 : a0b1c1 : a1b0c0 : a1b0c1 : a1b1c0 : a1b1c1]. The state ψ is separable as A B C if and only if its associated point [ψ] P7 lies in the generalized| i Segre variety given⊗ by⊗ ∈

ΣA B C := Im(fA B C). ⊗ ⊗ ⊗ ⊗ The above generalized Segre embedding factors in two equivalent ways:

fA B C = fA BC (IA fB+C )= fAB C (fA B IC ), ⊗ ⊗ ⊗ ◦ × ⊗ ◦ ⊗ × so we have a commutative square

fA⊗B IC P1 P1 P1 × / P3 P1 . A × B × C AB × C

IA fB+C fAB⊗C ×   f P1 P3 A⊗BC / P7 A × BC ABC

We will show (Theorem III.6) that the Segre variety ΣA B C agrees with the intersection ⊗ ⊗

ΣA B C =ΣA BC ΣAB C. ⊗ ⊗ ⊗ ∩ ⊗ In particular, we have

A BC (ψ)=0 and AB C(ψ)=0 [ψ] ΣA B C . J ⊗ J ⊗ ⇐⇒ ∈ ⊗ ⊗ 8

In summary, the observables AB C and A BC determine separability of any three particle state in the ABC order.J Of course,⊗ oneJ can⊗ also measure separability for the orders BAC and CAB by consistently taking into account permutations of the chosen Hilbert bases, as we next illustrate. Consider the following well-known states and their corresponding points in P7:

Sep = 000 [Sep] = [1 : 0 : 0 : 0 : 0 : 0 : 0 : 0] | i | i B = 1 ( 000 + 011 )[B ] = [1 : 0 : 0 : 1 : 0 : 0 : 0 : 0] | 1i √2 | i | i 1 B = 1 ( 000 + 101 )[B ] = [1 : 0 : 0 : 0 : 0 : 1 : 0 : 0] | 2i √2 | i | i 2 B = 1 ( 000 + 110 )[B ] = [1 : 0 : 0 : 0 : 0 : 0 : 1 : 0] | 3i √2 | i | i 3 W = 1 ( 100 + 010 + 001 )[W ] = [0 : 1 : 1 : 0 : 1 : 0 : 0 : 0] | i √3 | i | i | i GHZ = 1 ( 000 + 111 ) [GHZ] = [1 : 0 : 0 : 0 : 0 : 0 : 0 : 1] | i √2 | i | i These can be taken as representatives for the six existing equivalence classes of three particle states under SLOCC-equivalence. The state Sep is obviously separable and GHZ and W are the only genuinely entangled states. Moreover,| i these two entangled states| representi | twoi different equivalence classes of entanglement [17]. The former state, named after [18], is maximally entangled with respect to most entanglement measures existing in the literature and its one-particle reduced density matrices are all maximally mixed. The remaining states Bi are 2-partite (depending on the order of the Hilbert basis). We have the following table: | i

ABC Sep B1 B2 B3 W GHZ 8 A BC 0 1 1 0 1 J ⊗ 9 8 AB C 0 0 1 1 1 J ⊗ 9 0 1 1 1 8 1 J 2 2 9 Here the labels ABC indicate we are measuring entanglement of the states with the fixed order ABC and 1 := ( A BC + AB C ) J 2 J ⊗ J ⊗ is the average measure. In particular, we see that while B1 and B3 are bipartite in this order, the state B2 is classified as entangled. Note however that if we measure entanglement with respect to the order ACB, the roles of B2 and B3 are exchanged and we obtain the following table.

ACB Sep B1 B2 B3 W GHZ 8 A BC 0 1 0 1 1 J ⊗ 9 8 AB C 0 0 1 1 1 J ⊗ 9 0 1 1 1 8 1 J 2 2 9 The states Sep , W and GHZ are invariant under permutations of the basis ABC and so the values| of thei | observablesi | alwaysi remain unchanged. Note as well that W exhibits less entanglement than GHZ with respect to the above measures, in agreement| i with the existing entanglement measures.| i 9

III. HYPERCUBE OF SEGRE EMBEDDINGS

This section is purely mathematical. Given an integer n 2, we consider the generalized Segre embedding ≥

1 (n) 1 2n 1 P P P − × ···× −→ 2n 1 and introduce the notion of q-decomposability of a point z P − for any integer 1 < q n. We show that q-decomposability is detected by looking at∈ all the Segre embeddings of≤ the type

2ℓ 1 2n−ℓ 1 2n 1 P − P − P − , for 1 ℓ n 1, × −→ ≤ ≤ − which accommodate as edges of a directed hypercube of Segre embeddings. Let us first review some basic definitions and constructions. The complex projective space PN is the set of lines in the complex space CN+1 passing through the origin. It may be described as the quotient

CN+1 N 0 P := −{ }, λ C∗. z λz ∈ ∼ N A point z P will be denoted by its z = [z : : zN ] where, by ∈ 0 ··· definition, there is always at least an integer i such that zi =0, and for any λ C∗ we have 6 ∈

[z : : zN ] = [λz : : λzN ]. 0 ··· 0 ···

Definition III.1. Given positive integers k and ℓ, the Segre embedding fk,ℓ is the map

k ℓ (k+1)(ℓ+1) 1 fk,ℓ : P P P − × −→ k ℓ defined by sending a pair of points a = [a0 : : ak] P and b = [b0 : : bℓ] P to (k+1)(ℓ+1) 1 ··· ∈ ··· ∈ the point of P − whose homogeneous coordinates are the pairwise products of the homogeneous coordinates of a and b:

fk,ℓ(a,b) = [ : zij : ] with zij := aibj , ··· ··· where we take the lexicographical order.

The Segre embedding is injective, but not surjective in general. The image of fk,ℓ is called the Segre variety and is denoted by

(k+1)(ℓ+1) 1 Σk,ℓ := Im(fk,ℓ)= [ : zij : ] P − ; zij zi′j′ zij′ zi′j =0, i = i′j = j′ . ··· ··· ∈ − ∀ 6 6 n o In other words, Σk,ℓ is given by the zero locus of all the 2 2 minors of the matrix ×

z00 z0ℓ . ···. .  . .. .  . zk zkℓ  0 ···    A combinatorial argument shows that there is a total of

k +1 ℓ +1 k (k + 1) ℓ (ℓ + 1) ξk,ℓ := = · · · 2 · 2 4     10 minors of size 2 2 in such a matrix. × The above construction generalizes to products of more than two projective spaces of arbitrary dimensions as follows. Given positive integers k , , kn, let 1 ···

N(k , , kn) := (k + 1) (kn + 1) 1. 1 ··· 1 ··· − j j For 1 j n, let [a : : a ] denote coordinates of Pkj . ≤ ≤ 0 ··· kj Definition III.2. The generalized Segre embedding

Pk1 Pkn PN(k1, ,kn) fk , ,kn : ··· 1 ··· ×···× −→ is defined by letting

1 n 1 n fk1, ,kn ([ : ai : ], , [ : ai : ]) := [ : zi1 in : ] where zi1 in = ai ai ··· ··· 1 ··· ··· ··· n ··· ··· ··· ··· ··· 1 ··· n and the lexicographical is assumed. Denote the generalized Segre variety by

Σk , ,kn := Im(fk , ,kn ). 1 ··· 1 ··· It follows from the definition that every generalized Segre embedding may be written as compositions of maps of the form

Im fk,ℓ Im′ × × m for certain values of m, k, ℓ and m′, where Im denotes the identity map of P . These compositions may be arranged in a directed (n 1)-dimensional hypercube, where the initial k kn − N(k , ,kn) vertex is P 1 P and the final vertex is P 1 ··· . Note that the (n 1) final ×···× N(k , ,kn) − edges of the hypercube (those edges whose target is the final vertex P 1 ··· ) are given by Segre embeddings of bipartite-type

PN(k1, ,kj ) PN(kj+1, ,kn) PN(k1, ,kn) fN(k , ,kj ),N(kj , ,kn) : ··· ··· ··· , 1 ··· +1 ··· × −→ where 1 j n 1. ≤ ≤ − Example III.3 (4-particle states). We have already seen the examples of two and three particle states in the warm-up section. For the generalized Segre embedding f1,1,1,1 charac- 1 terizing entanglement of four particle states of spin- 2 , we obtain a cube with commutative faces

I I f1,1 P1 P1 P1 P1 × × / P1 P1 P3 . × × ❘×❘ × ×◆◆ ❘❘ I f I ◆◆ I f ❘❘❘ 1,1 ◆◆ 1,3 ×❘❘❘× ×◆◆ ❘❘❘) ◆◆◆' ) I f ' 1 3 1 3,1 / 1 7 f1,1 I I P P P × / P P × × × × ×

f1,1 I f1,3 I ×  ×   I f  3 1 1 1,1 / 3 3 P P P × / P P f1,7 × ×❘❘❘ × ◆◆ ❘❘ f , I ◆◆ f , ❘❘❘3 1× ◆◆3 3 ❘❘❘ ◆◆ ❘❘❘)  ◆◆◆  ) f7,1 ◆' P7 P1 / P15 × 11

We next state a general decomposability result for arbitrary products of projective spaces. 1 Since our interest lies in spin- 2 particle systems, for the sake of simplicity we will restrict to the case where the initial spaces are projective lines. For any integer m 1, let ≥ (m) m Nm := N(1, , 1)=2 1. ··· − We will consider the decompositions associated to the generalized Segre embedding

(n) P1 P1 PNn . × ···× −→ Definition III.4. Let n 2 and 1 < q n be integers. We will say that a point z PNn is ≥ ≤ ∈ q-decomposable if and only if there exist positive integers m1, ,mq with m1 + +mq = n such that ··· ···

z ΣNm , ,Nm . ∈ 1 ··· q Note that q-decomposable implies (q 1)-decomposable. If z is not 2-decomposable, we will say that it is indecomposable. − Points that are q-decomposable will correspond precisely to q-partite states and indecom- posable points will correspond to entangled states. Example III.5 (2- and 3-particle states). A point z in P3 is 2-decomposable if and only if 7 z Σ , . Otherwise it is indecomposable. A point z in P is 2-decomposable if and only if ∈ 1 1 z Σ , Σ , . It is 3-decomposable if and only if z Σ , , . The following result shows ∈ 3 1 ∪ 1 3 ∈ 1 1 1 that z is actually 3-decomposable if and only if z Σ3,1 Σ1,3, so that it is 2-decomposable in every possible way. In physical terms, it just says∈ that∩ a state is 3-partite if and only if it is 2-partite when considering both types of bipartitions. Theorem III.6 (Generalized Decomposability). Let n 2 and 1 < q n be integers. A point z PNn is q-decomposable if and only if it lies in at≥ least q 1 different≤ Segre varieties ∈ − of the form ΣN ,N , with 1 ℓ n 1. ℓ n−ℓ ≤ ≤ − For the particular extreme cases we have that a point z PNn is: ∈

Indecomposable z / ΣN ,N , for all 1 ℓ n 1. ⇐⇒ ∈ ℓ n−ℓ ≤ ≤ − ( n-decomposable z ΣNℓ,Nn−ℓ , for all 1 ℓ n 1. ⇐⇒ ∈ ≤ ≤ − We refer to the Appendix for the proof. This result will be essential in the next section, where we give a general method for measuring decomposability. Indeed, Theorem III.6 asserts that decomposability is entirely determined by the Segre varieties ΣNℓ,Nn−ℓ for all 1 ℓ n 1. Note that this family of varieties is the one arising when looking at the (n 1) edges≤ ≤ whose− target is the last vertex of the (n 1)-dimensional hypercube and corresponds− precisely to the family of all possible bipartitions− of PNn . This result will translate into taking (n 1) measures of a given n-particle state, in order to detect the level of decomposability of its− associated point in the projective space. Example III.7 (4-particle states). In the situation of Example III.3 and in view of Theorem III.6, a point z in P15 is:

indecomposable z / Σ , Σ , Σ , ⇐⇒ ∈ 7 1 ∪ 1 7 ∪ 3 3  2-decomposable z Σ7,1 Σ1,7 Σ3,3  ⇐⇒ ∈ ∪ ∪  3-decomposable z (Σ1,7 Σ7,1) (Σ1,7 Σ3,3) (Σ7,1 Σ3,3).  ⇐⇒ ∈ ∩ ∪ ∩ ∪ ∩ 4-decomposable z Σ7,1 Σ1,7 Σ3,3  ⇐⇒ ∈ ∩ ∩   12

IV. OPERATORS CONTROLLING EDGES OF THE HYPERCUBE

Given an n-particle state, in this section we define (n 1) observables which measure its entanglement. Let us first fix some notation. We will denote− by 1 0 0 1 0 i 1 0 σ = ; σ = ; σ = ; σ = 0 0 1 1 1 0 2 i −0 3 0 1        −  and by Ik the identity matrix of size k. The Kronecker product of two matrices = (aij ) Matk k and Matn n is the matrix of size kn kn given by: A ∈ × B ∈ × × a a k 11B··· 1 B := . . . . A⊗B  . . .  ak akk  1B··· B   The Hermitian product of two complex vectors u = (u , ,uk) and v = (v , , vk) is 0 ··· 0 ··· k u v = uivi. · i=0 X Also, let

k 2 u := u u = uiui. || || · i=0 X For α a complex number, we denote α := α = √α α its absolute value. We will use the Lagrange identity,| which| || states|| that·

k 1 k 2 2 2 − 2 u v u v = uivj uj vi . || || · || || − | · | | − | i=0 j=i+1 X X Note that in many references, the term on the right side of the identity is often written in 2 2 the equivalent form uivj uj vi instead of uivj uj vi . | − | | − | n In order to describe n-particle states we fix an ordered basis of Nn = 2 1 linearly independent vectors of the corresponding Hilbert space − i i = i i 1 n 1 n n | ··· i | iO1 ⊗ · · · ⊗ | iO where 1, , n denote the different observers and i1, ,in 0, 1 , since we are in O 1 ··· O { ··· }∈{ } the spin- 2 case. Using this basis, the coordinates for a general pure state for n particles will be written as

z0 z1   ψ n = . , | iO1···O .   zNn      where zi are complex numbers satisfying the normalization condition

2 zi =1. | | i 0 X≥ 13

Likewise, we will write:

ψ = (z , z , , zN ). h | 0 1 ··· n Given such a state, we may consider its class [ψ] PNn by taking its homogeneous coordinates ∈

[ψ] = [z : : zN ], 0 ··· n C where we recall that [z0 : : zNn ] = [λz0 : : λzNn ] for any λ ∗. For all ℓ =1, ,n 1,··· define the following··· observable acting∈ on n-particle states: ··· − 3 1 2 I n−ℓ n,ℓ(ψ) := 2 ℓ 1 ψ σi1 σiℓ 2 ψ . J − 2 − | h | ⊗···⊗ ⊗ | i |  i , ,iℓ=0 1 ···X   The purpose of this section is to show that n,ℓ(ψ)=0 if and only if the class [ψ] in the PNn J projective space lies in the Segre variety ΣNℓ,Nn−ℓ . The next two examples detail the computations of the observables introduced in Section II for the cases of two and three particles respectively. Note that we used a slightly different notation, namely:

2 particles: A B 2,1 and ΣA B Σ1,1. J ⊗ ≡ J ⊗ ≡ 3 particles: A BC 3,1, ΣA BC Σ1,3, AB C 3,2, and ΣAB C Σ3,1. J ⊗ ≡ J ⊗ ≡ J ⊗ ≡ J ⊗ ≡ Example IV.1 (2-particle states). We have a single observable

3 2 , (ψ)=2 ψ σi I ψ . J2 1 − | h | ⊗ 2 | i | i=0 ! X Define vectors = (z ,z ) and = (z ,z ), so that ψ = ( , ). Then we have A0 0 1 A1 2 3 h | A0 A1

2 2 , (ψ) =2 + + + J2 1 − |A0·A0 A1·A1| |A0·A1 A1·A0| + 2 + 2 = |A0·A1 − A1·A0| |A0·A0 − A1·A1| =1 z z + z z + z z + z z 2 + z z + z z + z z + z z 2 + − | 0 0 1 1 2 2 3 3| | 0 2 1 3 2 0 3 1| + z z z z + z z + z z 2 + z z + z z z z z z 2 = |− 0 2 − 1 3 2 0 3 1| | 0 0 1 1 − 2 2 − 3 3| =1 z z + z z + z z + z z 2 + z z z z + z z + zz 2 + − | 0 2 1 3 2 0 3 1| |− 0 2 − 1 3 2 0 3 1| + z z+ z z z z z z 2 =4 z z z z 2 . | 0 0 1 1 − 2 2 − 3 3| | 0 3 − 1 2|  Therefore , (ψ)=0 if and only if [ψ] Σ , . J2 1 ∈ 1 1 Example IV.2 (3-particle states). In this case we have two observables

3 2 , (ψ) := 2 ψ σi I ψ and J3 1 − i=0 | h | ⊗ 4 | i |  1 3  2 , (ψ) := 2 P ψ σi σj I ψ . J3 2 − 2 i,j=0 | h | ⊗ ⊗ 2 | i |  P  Write z = ( , ) where = (z , ,z ) and = (z , ,z ). Then we have A0 A1 A0 0 ··· 3 A1 4 ··· 7 14

2 2 2 , (ψ)= 2 ( + + + + + + J3 1 − |A0A0 A1A1| |A0A0 A1A1| |A0A1 A1A0| + + 2 + 2)= |− A0A1 A1A0| |A0A0 − A1A1| =1 ( + 2 + + 2 + 2)= − |A0A1 A1A0| |− A0A1 A1A0| |A0A0 − A1A1| =4 z z z z 2 + z z z z 2 + z z z z 2+ | 0 5 − 1 4| | 0 6 − 2 4| | 0 7 − 3 4| + zz z z 2 + z z z z 2 + z z z z 2 . | 1 6 − 2 5| | 1 7 − 3 5| | 2 7 − 3 6|

Note that the numbers zizj z′ z′ correspond to the minors describing the zero locus of − j i Σ , . Indeed, this is determined by the vanishing of all the 2 2 minors of the matrix 1 3 × z z z z 0 1 2 3 . z4 z5 z6 z7  

Therefore , (ψ)=0 if and only if [ψ] Σ , . J3 1 ∈ 1 3 Likewise, writing z = ( 0, 1, 2, 3) with 0 = (z0,z1), 1 = (z2,z3), 2 = (z4,z5) and = (z ,z ) we easilyA obtainA A A A A A A3 6 7 2 2 2 , (ψ)=4 z z z z + z z z z + z z z z + J3 2 | 0 3 − 1 2| | 0 5 − 1 4| | 0 7 − 1 6| + z z z z 2 + z z z z 2 + z z z z 2 . | 2 5 − 3 4| | 2 7 − 3 6| | 4 7 − 5 6|

Note that the numbers zizj zj′ zi′ correspond to the minors describing the zero locus of Σ3,1. Indeed, this is determined by− the vanishing of all the 2 2 minors of the matrix ×

z0 z1 z z 2 3 . z4 z5 z6 z7   Therefore , (ψ)=0 if and only if [ψ] Σ , . J3 2 ∈ 3 1 Returning to the general setting, note that the measures n,ℓ(ψ) are related to the trace of the squared density matrix for a given partition of the system.J Indeed, the density matrix for any physical state is a Hermitian operator and therefore can be written in terms of σi as

I n−ℓ I ℓ ρ = ci1, il σi1 σil 2 + dil+1 in 2 σil+1 σin , ··· ⊗···⊗ ⊗ ··· ⊗ ⊗···⊗ i ,...in 1,2,3 1 X∈{ } where we write explicitly the partition of the system in A and B. In this notation, the reduced density matrix of the system A reads

n l ρA =2 − ci , il σi σil 1 ··· 1 ⊗···⊗ . We may now compute expectation values for our operators:

ψ σi σi I n−ℓ ψ = Tr (ρ (σi σi I n−ℓ )) , h | 1 ⊗···⊗ ℓ ⊗ 2 | i · 1 ⊗···⊗ l ⊗ 2 and using the identity

n Tr ((σi σi ) (σj σj ))=2 δi j δi j δi j 1 ⊗···⊗ n · 1 ⊗···⊗ n 1 1 2 2 ··· n n 15 we obtain

n I n−ℓ ci1, il =2 ψ σi1 σiℓ 2 ψ . ··· h | ⊗···⊗ ⊗ | i We now compute the trace of the squared density matrix:

2 2(n l) l 2 I n−ℓ TrAρA =2 − ci1, il cj1, jl Tr ((σi1 σin ) (σj1 σjn ))=2 ψ σi1 σiℓ 2 ψ ··· ··· ⊗···⊗ · ⊗···⊗ h | ⊗· · ·⊗ ⊗ | i which leads to the identity

2 n,ℓ(ψ)=2 1 Tr ρ . J − A A The main result of this section is the following: 

Theorem IV.3. Let ψ be a pure n-particle state and let 1 ℓ n 1. Then | i ≤ ≤ − 2 n,ℓ(ψ)=4 MI , J | | I X where the sum runs over all 2 2 minors MI determining the zero locus of ΣNℓ,Nn−ℓ . In particular, ×

n,ℓ(ψ)=0 [ψ] ΣN ,N . J ⇐⇒ ∈ ℓ n−ℓ

Proof. Write (z , ,zN ) = ( , , N ), where 0 ··· n A0 ··· A ℓ

j = (z , ,z ) A j(Nn−ℓ+1) ··· j(Nn−ℓ+1)+Nn−ℓ for j =0, ,Nℓ, so that each j is with Nn ℓ +1 components. With this notation, we have ··· A −

Nℓ 1 Nℓ − 2 2 2 n,ℓ(ψ)=4 ( j k j k ). J ||A || · ||A || − |A · A | j=0 k>j X X Applying the Lagrange identity we obtain

Nℓ 1 Nℓ Nℓ 1 Nℓ − − 2 n,ℓ(ψ)=4 j,s k,t j,t k,s , J |A · A − A · A | j=0 k>j s=0 t>s X X X X where i,j denotes the j-th component of the tuple i. It now suffices to note that the numbersA A

j,s k,t j,t k,s A A − A A correspond to the minors MI . Indeed, the zero locus of ΣNℓ,Nn−ℓ is determined by the vanishing of the 2 2 minors of the matrix ×

0,0 0,Nn−ℓ A. ··· A .  . .  . N , N ,N A ℓ 0 ··· A ℓ n−ℓ    16

V. PHYSICAL INTERPRETATION

n Given an integer n 1, we fix an ordered basis of Nn = 2 1 linearly independent ≥ 1 − vectors of the Hilbert space of pure n-particle states of spin- 2 i i = i i , 1 n n 1 n n | ··· iO1···O | iO1 ⊗ · · · ⊗ | iO where 1, , n denote the different observers. We will omit the labels of the observers, but oneO should··· O note that, in the following, the chosen basis always has the same fixed order unless stated otherwise. Definition V.1. Let 1 q n be an integer. An n-particle state ψ is said to be q-partite if it can be written as ≤ ≤ | i

ψ = ψ ψq , | i | 1i ⊗ · · · ⊗ | i where ψi are ni-particle states, with ni > 0 and n + + nq = n. | i 1 ··· 1-partite states are called entangled, while n-partite states are called separable. Physically, separable states are those that are uncorrelated. A product state can thus be easily prepared in a local way: each observer i produces the state ψi and the measurement outcomes for each observer do not depend onO the outcomes for the| otheri observers. A basic observation is that a state is separable if and only if it lies in the generalized Segre variety of Definition III.2 (see for instance [5]). Likewise, a state is q-partite if and only if its corresponding projective point on PNn lies in a Segre variety of the form

ΣNm , ,Nm 1 ··· q with m1, ,mq positive integers such that m1+ +mq = n. So q-partite states correspond geometrically··· to the q-decomposable points of Definition··· III.4. Combining the results of the previous two sections we have that an n-particle state ψ is q-partite if and only if there are | i q 1 indices ℓ1, ,ℓq 1 with 1 ℓi n 1 and ℓi = ℓj such that n,ℓi (ψ)=0. Indeed, from− Theorem III.6··· we− know that≤ψ is≤q-partite− if and6 only if its correspondingJ point [ψ] in N | i P n lies in at least q 1 different Segre varieties of the form ΣN ,N , with 1 ℓ n 1. − ℓ n−ℓ ≤ ≤ − Moreover, from Theorem IV.3 we know that [ψ] ΣNℓ,Nn−ℓ if and only if n,ℓ(ψ)=0. Note that, a priori, given an n-particle state,∈ one would have to take (nJ 1)! measures of bipartite type to completely determine its q-decomposability (namely, for− each possible bipartition, check further bipartitions recursively). The hypercube approach tells us that it suffices to take (n 1) measures, corresponding to the operators n,ℓ in order to determine completely its q-decomposability.− {J } In the remaining of the section, we study the behaviour of the observables n,ℓ in some particular cases of interest. We first introduce the average observable acting onJ n-particle states:

n 1 1 − (ψ) := n,ℓ(ψ). J n 1 J ℓ − X=1 The two- and three- particle states discussed in Section II which are invariant under per- mutations of the Hilbert basis ( Sep , EPS , GHS , W ) allow for natural generalizations to the n-particle case. We study| theiri | entanglementi | i measures.| i Note first that the separable n-particle state

(n) Sep := 0 0 | ni | ··· i 17 corresponds to the point in PNn given by [Sep ] = [1 : 0 : : 0] n ··· and so one easily verifies that (Sepn)=0. The Schrödinger n-particle stateJ is a superposition of two maximally distinct states

1 (n) (n) Sn := 0 0 + 1 1 . | i √2 | ··· i | ··· i   It generalizes the two-particle state EPS and the three-particle state GHZ and it corre- sponds to the point in PNn given by| i | i

[Sn] = [1 : 0 : : 0 : 1]. ··· One easily computes n,ℓ(Sn)=1 for all 1 ℓ n and so (Sn)=1. For n > 3, its not clear that the SchrödingerJ state exhibits≤ maximal≤ entanglementJ [19]. This observation agrees with our measures for n,ℓ, as we will see below. We now consider a generalizationJ of the W state for three particles, to the case of n- particles. For each fixed integer 0 k n, these states are constructed by adding all permutations of generators of the form≤ ≤

(k) (n k) 1 1 0 − 0 | i⊗ · · · ⊗ | i ⊗ | i⊗ · · · ⊗ | i with k states 1 and n k states 0 , together with a global normalization constant. These are clearly invariant| i with− respect to| i permutations of the basis. Denote such states by

1 n − 2 (k) (n k) − Dn,k = 1 1 0 0 . | i k | i⊗ · · · ⊗ | i ⊗ | i⊗ · · · ⊗ | i permut   X These are known as Dicke states [20]. Note that Dn,0 = Sepn and D3,1 = W3 . In the case of four particles we have | i | i | i | i

1 D , = ( 1000 + 0100 + 0010 + 0001 ) , | 4 1i √4 | i | i | i | i 1 D , = ( 1100 + 1010 + 1001 + 0110 + 0101 + 0011 ) , | 4 2i √6 | i | i | i | i | i | i 1 D , = ( 1110 + 1101 + 1011 + 0111 ) . | 4 3i √4 | i | i | i | i Their corresponding points in P15 are

D , =[0:1:1:0:1:0:0:0:1:0:0:0:0:0:0:0], | 4 1i D , =[0:0:0:1:0:1:1:0:0:1:1:0:1:0:0:0], | 4 2i D , =[0:0:0:0:0:0:0:1:0:0:0:1:0:1:1:0]. | 4 3i We obtain the following table for the observables ,ℓ: J4

D , D , D , | 4 1i | 4 2i | 4 3i 3 3 , 1 J4 1 4 4 , 1 1 1 J4 2 3 3 , 1 J4 3 4 4 5 1 5 J 6 6 18

Note (D , )=1 as is the case for the state S . For more than four particles we obtain J 4 2 | 4i values > 1 for the observables n,ℓ. For instance, in the five particle case, we have: J

D , D , D , D , | 5 1i | 5 2i | 5 3i | 5 4i 16 24 24 16 , J5 1 25 25 25 25 24 27 27 24 , J5 2 25 25 25 25 . 24 27 27 24 , J5 3 25 25 25 25 16 24 24 16 , J5 4 25 25 25 25 4 51 51 4 J 5 50 50 5 In particular, we see that

(D , )= (D , ) > (S )=1. J 5 2 J 5 3 J 5 For higher particle states the same pattern repeats itself, with the middle states

Dn, n = Dn, n | ⌊ 2 ⌋i | ⌈ 2 ⌉i always exhibiting the largest entanglement as well as symmetries of the tables in both directions. We end this section with some notable examples in the four- and five-particle cases. The state 1 HS = 1100 + 0011 + ω 1001 + ω 0110 + ω2 1010 + ω2 0101 | i √6 | i | i | i | i | i | i

2πi  where ω = e 3 is a third root of unity, was conjectured to be maximally entangled by Higuchi-Sudbery [19] and it actually gives a local maximum of the average two-particle von Neumann entanglement entropy [21]. Another highly (though not maximally) entangled state, found by Brown-Stepney-Sudbery-Braunstein [22], is given by 1 BSSB = ( 0000 + + 011 + 1101 + 110 ) , | 4i 2 | i | i ⊗ | i | i |−i ⊗ | i where + = 1 ( 0 + 1 ) and = 1 ( 0 1 ). Our measures agree with these facts, as | i √2 | i | i |−i √2 | i − | i shown in the table below.

S D , BSSB HS | 4i | 4 2i | 4i | i 3 , 1 1 1 J4 1 4 5 4 , 1 1 . J4 2 4 3 , 1 1 1 1 J4 3 1 1 1 10 J 9 In [23], a related measure of entanglement is introduced, based on vector lengths and the angles between vectors of certain coefficient matrices. While this measure is strongly related to concurrence and hence to the observables , their measure Eavg does not distinguish the states BSSB and HS . In contrast, we doJ find that | 4i | i 1= (BSSB ) < (HS). J 4 J 19

In [22], a highly entangled five-particle state is described as 1 BSSB = ( 000 Φ + 010 Ψ + 100 Φ+ + 111 Ψ+ ) , | i5 2 | i ⊗ | −i | i ⊗ | −i | i ⊗ | i | i ⊗ | i where Ψ = 00 11 and Φ = 01 10 . Our measures give: | ±i | i ± | i | ±i | i ± | i

S D , BSSB | 5i | 5 2i | 5i 24 , 1 1 J5 1 25 27 3 , 1 J5 2 25 2 27 5 , 1 J5 3 25 4 24 , 1 1 J5 4 25 1 51 19 J 50 16 In particular, we see that

1= (S ) < (D , ) < (BSSB ). J 5 J 5 2 J 5

VI. SUMMARY AND CONCLUSIONS

Within a purely geometric framework, we have described entanglement of n-particle states in terms of a hypercube of bipartite-type Segre maps. For simplicity, in this paper we have 1 restricted to spin- 2 particles, but the geometric results generalize almost verbatim to the case of qudits. The hypercube picture allows to identify separability (or more generally, q-decomposability) in terms of a depth factor within the hypercube: the deeper a state lies in the hypercube, the more separable. We have defined a collection of operators which measure the properties of a state in the above geometric set-up. Given an n-particle state and having fixed an ordered basis n 1 of the total Hilbert space, there are 2 − different decomposability possibilities, given by the different ordered q-partitions, for 1 q n. A standard way to characterize the decomposability of such a state would be≤ to consider,≤ for any possible bipartition, all of its possible bipartitions in a recursive way. This gives a total of (n 1)! measures to be taken. Our hypercube approach says that it suffices to take (n 1) measures,− given by the operators − n,ℓ, with 1 ℓ n 1. So as the complexity of the problem grows factorially, our solution justJ grows linearly≤ ≤ on−n. The operators n,ℓ measure the different bipartitions of the system, corresponding geo- metrically with theJ last (n 1) edges of the Segre hypercube. The expected value of these operators is related with the− quantum Tsalis entropy (q =2) of both parts of the state. The concrete values of ℓ for which n,ℓ vanishes shows precisely in what edge of the hypercube the state belongs or, more physically,J in which parts the state is separable. To illustrate the physical interest of our approach, we have computed the value of the operators n,ℓ for various entangled states considered in the literature. The motivation for many of themJ arises in quantum computing and are therefore classified from the quantum control . In all cases, the proposed observables give results consistent with the expectations. 20

Appendix A: Proof of the Generalized Decomposability Theorem

This appendix is devoted to the proof of Theorem III.6 on geometric decomposability. Let us first consider the tripartite-type Segre embedding

k1 k2 k3 N(k1,k2,k3) fk ,k ,k : P P P P , 1 2 3 × × → where we recall that

N(k , , kn) := (k + 1) (kn + 1) 1. 1 ··· 1 ··· − The following lemma relates the Segre varieties associated to it.

Lemma A.1. Let k1, k2, k3 be positive integers. Then

Σk ,k ,k =Σ Σ . 1 2 3 k1,N(k2,k3) ∩ N(k1,k2),k3

Proof. It suffices to prove the inclusion Σk1,N(k2,k3) ΣN(k1,k2),k3 Σk1,k2,k3 . Recall that we have identities ∩ ⊆

fk ,k ,k = f (I fk ,k )= f (fk ,k I). 1 2 3 k1,N(k2,k3) ◦ × 2 3 N(k1,k2),k3 ◦ 1 2 × k k k Let a = [ai], b = [bj ] and c = [ck] be coordinates for P 1 , P 2 and P 3 respectively. We will N(k ,k ) N(k ,k ) also let x = [xij ] and y = [yjk] be coordinates for P 1 2 and P 2 3 respectively, and N(k ,k ,k ) z = [zijk] will denote coordinates for P 1 2 3 . We have:

(fk ,k I)(a,b,c) = (x, c), with xij := aibj. 1 2 × (I fk ,k )(a,b,c) = (a,y), with yjk := bjck. × 2 3 fN(k1,k2),k3 (x, c) = (z), with zijk := xij ck.

fk1,N(k2,k3)(a,y) = (z), with zijk := aiyjk.

N(k ,k ,k ) Given a point z = (zijk) P 1 2 3 , we claim the following: ∈ 1. z Σ if and only if all the 2 2 minors of the matrix ∈ k1,N(k2,k3) ×

z000 z0k2k3 . ··· .  . .  zk zk k k  100 ··· 1 2 3    vanish, so that zijk zi′j′k′ = zi′jk zij′ k′ for all i = i′ and (j, k) = (j′, k′). · · 6 6 2. z Σ if and only if all the 2 2 minors of the matrix ∈ N(k1,k2),k3 ×

z000 z00k3 . ··· .  . .  zk k zk k k  1 20 ··· 1 2 3    vanish, so that zijk zi′j′k′ = zijk′ zi′j′k for all (i, j) = (i′, j′) and k = k′. · · 6 6 21

3. z Σk ,k ,k if and only if z Σ , so that zijk = xij ck, and x = (xij ) ∈ 1 2 3 ∈ N(k1,k2),k3 · ∈ Σk ,k . This last condition gives the vanishing of the 2 2 minors of the matrix 1 2 ×

x00 x0k2 . ··· .  . .  . xk xk k  10 ··· 1 2    Therefore we have xij xi′j′ = xij′ xi′j for all i = i′, j = j′. This gives identities · · 6 6

zijk zi′j′k′ = xij ck xi′j′ ck′ = xi′ j ck xij′ ck′ = zi′jk zij′k′ · · · · · · · ·

for all i = i′, j = j′ and k = k′. 6 6 6 Claims (1) and (2) are straightforward, while (3) follows from the identity

Σk ,k ,k = Im(f (fk ,k I)). 1 2 3 N(k1,k2),k3 ◦ 1 2 × Assume now that z Σ Σ . Then the equations in (1) and (2) are ∈ k1,N(k2,k3) ∩ N(k1,k2),k3 satisfied, and moreover we may write zijk = xij ck. To show that z Σk ,k ,k it only · ∈ 1 2 3 remains to prove that xij xi′ j′ = xij′ xi′ j for all i = i′ and j = j′. By (1) we have · · 6 6

zijk zi′j′k′ = xij ck xi′j′ ck′ = xi′j ck xij′ ck′ = zi′jk zij′ k′ · · · · · · · · for all i = i′ and (j, k) = (j′, k′). Take k = k′ such that ck =0. This gives 6 6 6 2 2 c xij xi′j′ = c xi′j xij′ for all i = i′ and j = j′. k · · k · · 6 6 Since c2 =0, we obtain the desired identities. k 6 In order to prove the Generalized Decomposability Theorem it will be useful to denote the cubical decomposition of

(n) P1 P1 PNn × ···× −→ by the (n 1)-dimensional hypercube whose vertices are given by tuples v = (v1, , vn 1) − (n) ··· − 1 1 where vi 0, 1 , with initial vertex v0 = (0, , 0) representing P P and final ∈ { } Nn ··· × ··· × vertex vf = (1, , 1) representing P . We will call v := v1 + + vn 1 the degree of a vertex. All edges··· are of the form | | ··· −

[j] (v1, , vn 1) (w1, , wn 1) ··· − −→ ··· − where vi = wi for all i = j and wj = vj +1=1, so that w = v +1. Such an edge represents a contraction of6 a product at the position j via a| Segre| | embedding.| × Example A.2. For example, the cubical representation of f1,1,1 is

[1] (00) / (10) ,

[2] [2]    [1]  (01) / (11) 22 and the cubical representation of f1,1,1,1 is

[3] (000) / (001) . ●● ●● ●●[2] ●●[2] ●● ●● ●●# ●●# [3] / [1] (010) / (011)

[1]  [1]  [3] / (100) / (101) [1] ●● ●● ●●[2] ●●[2] ●● ●● ●●#  ●●#  #  [3] #  (110) / (111)

Nn We will say that a point z P lives in a vertex v = (v1, , vn 1) if it is in the image ∈ ··· − of the map v vf given by the composition of all edges [i] such that vi =0. It follows that z is q-decomposable→ if and only if it lives in some vertex v of degree v = n q. | | − Lemma A.3. Let v = v′ be two vertices of the same degree in the (n 1)-dimensional hypercube. Then there6 is a unique 2-dimensional face of the form −

[i] / w′ v′

[j] [j]  [i]  v / w and if z lives in w, v and v′, then it also lives in w′.

Proof. Since v = v′ and v = v′, there are i, j such that vi =0, v′ =1, vj =1, v′ =0, and | | | | 6 i j vk = v′ for all k = i, j. Let w and w′ be the vertices whose components are given by k 6 wk = max vk, v′ and w′ = min vk, v′ . { k} k { k} This gives the above commutative square. Assume now that z lives in w, v and v′. It suffices to consider two cases: Case j = i +1. In this case, the above commutative square represents of the form

I f I I A k1,k2 k3 B / A Pk1 Pk2 Pk3 B × × × / A PN(k1,k2) Pk3 B × × × × × × ×

IA Ik fk ,k IB IA fN k ,k ,k IB  × 1 × 2 3 ×  × ( 1 2) 3 ×  I f I  A k1,N(k2,k3) B / A Pk1 PN(k2,k3) B × × / A PN(k1,k2,k3) B, × × × × × where A and B are products of projective spaces. Therefore we can apply Lemma A.1 to conclude that z lives in w′. Case j

Since z lives in w we may decompose z = (zA,z12,zB,z34,zC). Since it lives in v, we have z Σ and since it lives in v′ we have z Σ . It directly follows that z 12 ∈ N(k1,k2) 34 ∈ N(k3,k4) lives in w′. Theorem A.4 (Generalized Decomposability). Let n 2 and 1 < q n be integers. A point z PNn is q-decomposable if and only if it is in at≥ least q 1 different≤ Segre varieties ∈ − of the form ΣN ,N , with 0 ℓ n. ℓ n−ℓ ≤ ≤ Proof. Using the cubical representation introduced above, it suffices to show that if z lives 1 q 1 i in q 1 different vertices v , , v − of degree v = n 2, then z lives in a vertex of degree − ··· | | − (n q). By recursively applying Lemma A.3 we find that z lives in the vertex v∗ of degree − i (n q) whose components are vj∗ = mini vj . This implies that z is q-decomposable. The converse− statement is trivial. { }

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